L. Nirenberg
Courant Institute

A Geometric Problem and the Hopf Lemma

Abstract: Let $U$ be a bounded convex body in $\mathbb{R}_n$ with smooth boundary $M$. Assume that for any two points $(x',a)$ and $(x',b)$ on $M$, with $a<b$ the mean curvature of $M$ at the first is not less than that at the second. Under some additional condition we show that $M$ is symmetric about a hyperplane $x_n$ = constant. Relations to generalized Hopf Lemma are discussed.